The End of an Era

The mathematics REU is officially over! After two nights of talks and good (and free) food and a short party, it ended. I know much more mathematics, and I continued my ongoing trend of feeling less and less intelligent. I'm also $1500, minus about $350 of tax, wealthier. So it was basically a fantastic time. I could recite an endless string of gripping results and definitions, but I'd rather mention one brief "life lesson" that I found to be pretty important.

There is a general trend particularly in physics, and to a lesser extent in mathematics, in which higher levels of abstraction and generality are considered more attractive, profound, and interesting. There are legitimate reasons why this is so. Abstract notions explain or are applicable to a wider range of particular situations. Physicists usually call this feature "profundity". They are absolutely correct: consider Maxwell's equations, for example. These four short equations express all the features of classical electromagnetism. If that isn't profound, useful, and possessing a high degree of explanatory power, then I don't know what is.

But the observation that generality tends to be better often morphs into the belief that generality is better. Under this illusion, I dragged myself into the study of category theory. Category theory is so general that it is practically meaningless by itself. I'm not quite sure how to describe this, but it just is. This is not to say that category theory is useless. On the contrary, it is used everywhere, but mostly as a sort of language rather than a piece of actual mathematics. Moreover, it's applications are primarily in algebra, of which I have little knowledge. So I spent my recent weeks regretting my choice as I labored writing my paper on categories, regretting my mistaken assumption that generality for its own sake has any value whatsoever.